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Properties

Label 2873.2.a.u
Level $2873$
Weight $2$
Character orbit 2873.a
Self dual yes
Analytic conductor $22.941$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2873,2,Mod(1,2873)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2873.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2873, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2873 = 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2873.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-8,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.9410205007\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 13x^{10} + 57x^{8} - 104x^{6} + 78x^{4} - 19x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 221)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + \beta_{2} q^{4} + ( - \beta_{10} + \beta_{3}) q^{5} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_1) q^{6} + (\beta_{7} - \beta_{6}) q^{7} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{8}+ \cdots + ( - 5 \beta_{11} + 4 \beta_{10} + \cdots - 11 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{3} + 2 q^{4} + 8 q^{9} - 6 q^{10} - 20 q^{12} + 10 q^{14} + 2 q^{16} + 12 q^{17} - 22 q^{22} - 24 q^{23} - 22 q^{25} - 20 q^{27} - 22 q^{29} - 16 q^{30} - 8 q^{35} + 46 q^{36} - 12 q^{38} + 34 q^{40}+ \cdots - 26 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 13x^{10} + 57x^{8} - 104x^{6} + 78x^{4} - 19x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 10\nu^{9} + 22\nu^{7} + 22\nu^{5} - 81\nu^{3} + 38\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} - 25\nu^{8} + 104\nu^{6} - 181\nu^{4} + 133\nu^{2} - 24 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{10} - 50\nu^{8} + 203\nu^{6} - 317\nu^{4} + 171\nu^{2} - 13 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{11} - 50\nu^{9} + 203\nu^{7} - 317\nu^{5} + 176\nu^{3} - 33\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{11} + 40\nu^{9} - 181\nu^{7} + 334\nu^{5} - 217\nu^{3} + 11\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{10} - 40\nu^{8} + 181\nu^{6} - 334\nu^{4} + 217\nu^{2} - 16 ) / 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{10} - 85\nu^{8} + 329\nu^{6} - 476\nu^{4} + 228\nu^{2} - 19 ) / 5 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\nu^{11} + 13\nu^{9} - 57\nu^{7} + 104\nu^{5} - 78\nu^{3} + 19\nu \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -13\nu^{11} + 165\nu^{9} - 691\nu^{7} + 1149\nu^{5} - 697\nu^{3} + 76\nu ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - 2\beta_{5} - \beta_{4} + 6\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} - 8\beta_{10} - 9\beta_{7} + 7\beta_{6} + 9\beta_{3} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 9\beta_{8} - 19\beta_{5} - 7\beta_{4} + 35\beta_{2} + 50 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 54\beta_{11} - 56\beta_{10} - 63\beta_{7} + 42\beta_{6} + 65\beta_{3} + 165\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 65\beta_{9} + 63\beta_{8} - 140\beta_{5} - 42\beta_{4} + 207\beta_{2} + 297 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 347\beta_{11} - 370\beta_{10} - 410\beta_{7} + 249\beta_{6} + 435\beta_{3} + 995\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 435\beta_{9} + 410\beta_{8} - 943\beta_{5} - 249\beta_{4} + 1244\beta_{2} + 1806 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2187\beta_{11} - 2373\beta_{10} - 2597\beta_{7} + 1493\beta_{6} + 2808\beta_{3} + 6071\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.48782
−1.85923
−1.35007
−1.07190
−0.557160
−0.268136
0.268136
0.557160
1.07190
1.35007
1.85923
2.48782
−2.48782 −3.07819 4.18927 −1.33009 7.65798 0.793257 −5.44652 6.47523 3.30904
1.2 −1.85923 1.06781 1.45675 2.26991 −1.98530 0.512821 1.01004 −1.85979 −4.22029
1.3 −1.35007 −0.321155 −0.177318 −0.991347 0.433581 −4.77812 2.93953 −2.89686 1.33839
1.4 −1.07190 −2.24547 −0.851021 2.66497 2.40693 −2.62488 3.05602 2.04214 −2.85659
1.5 −0.557160 2.03103 −1.68957 0.0627664 −1.13161 2.89007 2.05568 1.12509 −0.0349709
1.6 −0.268136 −1.45403 −1.92810 1.99740 0.389877 −1.01727 1.05327 −0.885808 −0.535575
1.7 0.268136 −1.45403 −1.92810 −1.99740 −0.389877 1.01727 −1.05327 −0.885808 −0.535575
1.8 0.557160 2.03103 −1.68957 −0.0627664 1.13161 −2.89007 −2.05568 1.12509 −0.0349709
1.9 1.07190 −2.24547 −0.851021 −2.66497 −2.40693 2.62488 −3.05602 2.04214 −2.85659
1.10 1.35007 −0.321155 −0.177318 0.991347 −0.433581 4.77812 −2.93953 −2.89686 1.33839
1.11 1.85923 1.06781 1.45675 −2.26991 1.98530 −0.512821 −1.01004 −1.85979 −4.22029
1.12 2.48782 −3.07819 4.18927 1.33009 −7.65798 −0.793257 5.44652 6.47523 3.30904
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)
\(17\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2873.2.a.u 12
13.b even 2 1 inner 2873.2.a.u 12
13.f odd 12 2 221.2.m.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.m.b ✓ 12 13.f odd 12 2
2873.2.a.u 12 1.a even 1 1 trivial
2873.2.a.u 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2873))\):

\( T_{2}^{12} - 13T_{2}^{10} + 57T_{2}^{8} - 104T_{2}^{6} + 78T_{2}^{4} - 19T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{6} + 4T_{3}^{5} - 3T_{3}^{4} - 22T_{3}^{3} - 6T_{3}^{2} + 22T_{3} + 7 \) Copy content Toggle raw display
\( T_{5}^{12} - 19T_{5}^{10} + 132T_{5}^{8} - 410T_{5}^{6} + 552T_{5}^{4} - 256T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 13 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} + 4 T^{5} - 3 T^{4} + \cdots + 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 19 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 40 T^{10} + \cdots + 225 \) Copy content Toggle raw display
$11$ \( T^{12} - 82 T^{10} + \cdots + 765625 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T - 1)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 99 T^{10} + \cdots + 5625 \) Copy content Toggle raw display
$23$ \( (T^{6} + 12 T^{5} + \cdots + 873)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 11 T^{5} + \cdots + 127947)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 87 T^{10} + \cdots + 3272481 \) Copy content Toggle raw display
$37$ \( T^{12} - 246 T^{10} + \cdots + 39125025 \) Copy content Toggle raw display
$41$ \( T^{12} - 115 T^{10} + \cdots + 16129 \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} - 101 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 749062161 \) Copy content Toggle raw display
$53$ \( (T^{6} - 3 T^{5} + \cdots - 65367)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 163 T^{10} + \cdots + 18769 \) Copy content Toggle raw display
$61$ \( (T^{6} + 13 T^{5} + \cdots + 11347)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 34735640625 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15006985009 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15430856841 \) Copy content Toggle raw display
$79$ \( (T^{6} + 19 T^{5} + \cdots + 16987)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 120 T^{10} + \cdots + 18769 \) Copy content Toggle raw display
$89$ \( T^{12} - 78 T^{10} + \cdots + 332929 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 939722788449 \) Copy content Toggle raw display
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